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Theorem itg1addlem4 23686
Description: Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
itg1add.3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
itg1add.4 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
Assertion
Ref Expression
itg1addlem4 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
Distinct variable groups:   𝑖,𝑗,𝑦,𝑧   𝑦,𝐼   𝑦,𝑃,𝑧   𝑖,𝐹,𝑗,𝑦,𝑧   𝑖,𝐺,𝑗,𝑦,𝑧   𝜑,𝑖,𝑗,𝑦,𝑧
Allowed substitution hints:   𝑃(𝑖,𝑗)   𝐼(𝑧,𝑖,𝑗)

Proof of Theorem itg1addlem4
Dummy variables 𝑤 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . 5 (𝜑𝐹 ∈ dom ∫1)
2 i1fadd.2 . . . . 5 (𝜑𝐺 ∈ dom ∫1)
31, 2i1fadd 23682 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom ∫1)
4 i1frn 23664 . . . . . . . 8 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
51, 4syl 17 . . . . . . 7 (𝜑 → ran 𝐹 ∈ Fin)
6 i1frn 23664 . . . . . . . 8 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
72, 6syl 17 . . . . . . 7 (𝜑 → ran 𝐺 ∈ Fin)
8 xpfi 8473 . . . . . . 7 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
95, 7, 8syl2anc 575 . . . . . 6 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
10 ax-addf 10303 . . . . . . . . . 10 + :(ℂ × ℂ)⟶ℂ
11 ffn 6259 . . . . . . . . . 10 ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ))
1210, 11ax-mp 5 . . . . . . . . 9 + Fn (ℂ × ℂ)
13 i1ff 23663 . . . . . . . . . . . . 13 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
141, 13syl 17 . . . . . . . . . . . 12 (𝜑𝐹:ℝ⟶ℝ)
1514frnd 6266 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ⊆ ℝ)
16 ax-resscn 10281 . . . . . . . . . . 11 ℝ ⊆ ℂ
1715, 16syl6ss 3817 . . . . . . . . . 10 (𝜑 → ran 𝐹 ⊆ ℂ)
18 i1ff 23663 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
192, 18syl 17 . . . . . . . . . . . 12 (𝜑𝐺:ℝ⟶ℝ)
2019frnd 6266 . . . . . . . . . . 11 (𝜑 → ran 𝐺 ⊆ ℝ)
2120, 16syl6ss 3817 . . . . . . . . . 10 (𝜑 → ran 𝐺 ⊆ ℂ)
22 xpss12 5333 . . . . . . . . . 10 ((ran 𝐹 ⊆ ℂ ∧ ran 𝐺 ⊆ ℂ) → (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ))
2317, 21, 22syl2anc 575 . . . . . . . . 9 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ))
24 fnssres 6218 . . . . . . . . 9 (( + Fn (ℂ × ℂ) ∧ (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ)) → ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
2512, 23, 24sylancr 577 . . . . . . . 8 (𝜑 → ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
26 itg1add.4 . . . . . . . . 9 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
2726fneq1i 6199 . . . . . . . 8 (𝑃 Fn (ran 𝐹 × ran 𝐺) ↔ ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
2825, 27sylibr 225 . . . . . . 7 (𝜑𝑃 Fn (ran 𝐹 × ran 𝐺))
29 dffn4 6340 . . . . . . 7 (𝑃 Fn (ran 𝐹 × ran 𝐺) ↔ 𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃)
3028, 29sylib 209 . . . . . 6 (𝜑𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃)
31 fofi 8494 . . . . . 6 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ 𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃) → ran 𝑃 ∈ Fin)
329, 30, 31syl2anc 575 . . . . 5 (𝜑 → ran 𝑃 ∈ Fin)
33 difss 3943 . . . . 5 (ran 𝑃 ∖ {0}) ⊆ ran 𝑃
34 ssfi 8422 . . . . 5 ((ran 𝑃 ∈ Fin ∧ (ran 𝑃 ∖ {0}) ⊆ ran 𝑃) → (ran 𝑃 ∖ {0}) ∈ Fin)
3532, 33, 34sylancl 576 . . . 4 (𝜑 → (ran 𝑃 ∖ {0}) ∈ Fin)
36 opelxpi 5355 . . . . . . . . . 10 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺))
37 ffun 6262 . . . . . . . . . . . 12 ( + :(ℂ × ℂ)⟶ℂ → Fun + )
3810, 37ax-mp 5 . . . . . . . . . . 11 Fun +
3910fdmi 6269 . . . . . . . . . . . 12 dom + = (ℂ × ℂ)
4023, 39syl6sseqr 3856 . . . . . . . . . . 11 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ dom + )
41 funfvima2 6721 . . . . . . . . . . 11 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
4238, 40, 41sylancr 577 . . . . . . . . . 10 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
4336, 42syl5 34 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
4443imp 395 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺)))
45 df-ov 6880 . . . . . . . 8 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
4626rneqi 5560 . . . . . . . . 9 ran 𝑃 = ran ( + ↾ (ran 𝐹 × ran 𝐺))
47 df-ima 5331 . . . . . . . . 9 ( + “ (ran 𝐹 × ran 𝐺)) = ran ( + ↾ (ran 𝐹 × ran 𝐺))
4846, 47eqtr4i 2838 . . . . . . . 8 ran 𝑃 = ( + “ (ran 𝐹 × ran 𝐺))
4944, 45, 483eltr4g 2909 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ ran 𝑃)
5014ffnd 6260 . . . . . . . 8 (𝜑𝐹 Fn ℝ)
51 dffn3 6270 . . . . . . . 8 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
5250, 51sylib 209 . . . . . . 7 (𝜑𝐹:ℝ⟶ran 𝐹)
5319ffnd 6260 . . . . . . . 8 (𝜑𝐺 Fn ℝ)
54 dffn3 6270 . . . . . . . 8 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
5553, 54sylib 209 . . . . . . 7 (𝜑𝐺:ℝ⟶ran 𝐺)
56 reex 10315 . . . . . . . 8 ℝ ∈ V
5756a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
58 inidm 4026 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
5949, 52, 55, 57, 57, 58off 7145 . . . . . 6 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶ran 𝑃)
6059frnd 6266 . . . . 5 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ran 𝑃)
6160ssdifd 3952 . . . 4 (𝜑 → (ran (𝐹𝑓 + 𝐺) ∖ {0}) ⊆ (ran 𝑃 ∖ {0}))
6215sselda 3805 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
6320sselda 3805 . . . . . . . . . 10 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
6462, 63anim12dan 607 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺)) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
65 readdcl 10307 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 + 𝑧) ∈ ℝ)
6664, 65syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺)) → (𝑦 + 𝑧) ∈ ℝ)
6766ralrimivva 3166 . . . . . . 7 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ)
68 funimassov 7044 . . . . . . . 8 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
6938, 40, 68sylancr 577 . . . . . . 7 (𝜑 → (( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
7067, 69mpbird 248 . . . . . 6 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ)
7148, 70syl5eqss 3853 . . . . 5 (𝜑 → ran 𝑃 ⊆ ℝ)
7271ssdifd 3952 . . . 4 (𝜑 → (ran 𝑃 ∖ {0}) ⊆ (ℝ ∖ {0}))
73 itg1val2 23671 . . . 4 (((𝐹𝑓 + 𝐺) ∈ dom ∫1 ∧ ((ran 𝑃 ∖ {0}) ∈ Fin ∧ (ran (𝐹𝑓 + 𝐺) ∖ {0}) ⊆ (ran 𝑃 ∖ {0}) ∧ (ran 𝑃 ∖ {0}) ⊆ (ℝ ∖ {0}))) → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))))
743, 35, 61, 72, 73syl13anc 1484 . . 3 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))))
7519adantr 468 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝐺:ℝ⟶ℝ)
767adantr 468 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → ran 𝐺 ∈ Fin)
77 inss2 4037 . . . . . . . . 9 ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
7877a1i 11 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
79 i1fima 23665 . . . . . . . . . . 11 (𝐹 ∈ dom ∫1 → (𝐹 “ {(𝑤𝑧)}) ∈ dom vol)
801, 79syl 17 . . . . . . . . . 10 (𝜑 → (𝐹 “ {(𝑤𝑧)}) ∈ dom vol)
81 i1fima 23665 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
822, 81syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
83 inmbl 23529 . . . . . . . . . 10 (((𝐹 “ {(𝑤𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8480, 82, 83syl2anc 575 . . . . . . . . 9 (𝜑 → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8584ad2antrr 708 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8633, 71syl5ss 3816 . . . . . . . . . . . . 13 (𝜑 → (ran 𝑃 ∖ {0}) ⊆ ℝ)
8786sselda 3805 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝑤 ∈ ℝ)
8887adantr 468 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ∈ ℝ)
8963adantlr 697 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
9088, 89resubcld 10746 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑤𝑧) ∈ ℝ)
9188recnd 10356 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ∈ ℂ)
9289recnd 10356 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
9391, 92npcand 10684 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧) + 𝑧) = 𝑤)
94 eldifsni 4519 . . . . . . . . . . . . 13 (𝑤 ∈ (ran 𝑃 ∖ {0}) → 𝑤 ≠ 0)
9594ad2antlr 709 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ≠ 0)
9693, 95eqnetrd 3052 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧) + 𝑧) ≠ 0)
97 oveq12 6886 . . . . . . . . . . . . 13 (((𝑤𝑧) = 0 ∧ 𝑧 = 0) → ((𝑤𝑧) + 𝑧) = (0 + 0))
98 00id 10499 . . . . . . . . . . . . 13 (0 + 0) = 0
9997, 98syl6eq 2863 . . . . . . . . . . . 12 (((𝑤𝑧) = 0 ∧ 𝑧 = 0) → ((𝑤𝑧) + 𝑧) = 0)
10099necon3ai 3010 . . . . . . . . . . 11 (((𝑤𝑧) + 𝑧) ≠ 0 → ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0))
10196, 100syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0))
102 itg1add.3 . . . . . . . . . . 11 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
1031, 2, 102itg1addlem3 23685 . . . . . . . . . 10 ((((𝑤𝑧) ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0)) → ((𝑤𝑧)𝐼𝑧) = (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
10490, 89, 101, 103syl21anc 857 . . . . . . . . 9 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) = (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
1051, 2, 102itg1addlem2 23684 . . . . . . . . . . 11 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
106105ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
107106, 90, 89fovrnd 7039 . . . . . . . . 9 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) ∈ ℝ)
108104, 107eqeltrrd 2893 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
10975, 76, 78, 85, 108itg1addlem1 23679 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
11087recnd 10356 . . . . . . . . 9 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝑤 ∈ ℂ)
1111, 2i1faddlem 23680 . . . . . . . . 9 ((𝜑𝑤 ∈ ℂ) → ((𝐹𝑓 + 𝐺) “ {𝑤}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})))
112110, 111syldan 581 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → ((𝐹𝑓 + 𝐺) “ {𝑤}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})))
113112fveq2d 6415 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘((𝐹𝑓 + 𝐺) “ {𝑤})) = (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
114104sumeq2dv 14659 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
115109, 113, 1143eqtr4d 2857 . . . . . 6 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘((𝐹𝑓 + 𝐺) “ {𝑤})) = Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧))
116115oveq2d 6893 . . . . 5 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))) = (𝑤 · Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧)))
117107recnd 10356 . . . . . 6 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) ∈ ℂ)
11876, 110, 117fsummulc2 14741 . . . . 5 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
119116, 118eqtrd 2847 . . . 4 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))) = Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
120119sumeq2dv 14659 . . 3 (𝜑 → Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
12191, 117mulcld 10348 . . . . 5 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
122121anasss 454 . . . 4 ((𝜑 ∧ (𝑤 ∈ (ran 𝑃 ∖ {0}) ∧ 𝑧 ∈ ran 𝐺)) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
12335, 7, 122fsumcom 14732 . . 3 (𝜑 → Σ𝑤 ∈ (ran 𝑃 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
12474, 120, 1233eqtrd 2851 . 2 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
125 oveq1 6884 . . . . . . 7 (𝑦 = (𝑤𝑧) → (𝑦 + 𝑧) = ((𝑤𝑧) + 𝑧))
126 oveq1 6884 . . . . . . 7 (𝑦 = (𝑤𝑧) → (𝑦𝐼𝑧) = ((𝑤𝑧)𝐼𝑧))
127125, 126oveq12d 6895 . . . . . 6 (𝑦 = (𝑤𝑧) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)))
12832adantr 468 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝑃 ∈ Fin)
12971adantr 468 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝑃 ⊆ ℝ)
130129sselda 3805 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑣 ∈ ℝ)
13163adantr 468 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑧 ∈ ℝ)
132130, 131resubcld 10746 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → (𝑣𝑧) ∈ ℝ)
133132ex 399 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 → (𝑣𝑧) ∈ ℝ))
134130recnd 10356 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑣 ∈ ℂ)
135134adantrr 699 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑣 ∈ ℂ)
13671sselda 3805 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ran 𝑃) → 𝑦 ∈ ℝ)
137136ad2ant2rl 746 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑦 ∈ ℝ)
138137recnd 10356 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑦 ∈ ℂ)
13963recnd 10356 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
140139adantr 468 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑧 ∈ ℂ)
141135, 138, 140subcan2ad 10725 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → ((𝑣𝑧) = (𝑦𝑧) ↔ 𝑣 = 𝑦))
142141ex 399 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → ((𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃) → ((𝑣𝑧) = (𝑦𝑧) ↔ 𝑣 = 𝑦)))
143133, 142dom2lem 8235 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ)
144 f1f1orn 6367 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
145143, 144syl 17 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
146 oveq1 6884 . . . . . . . 8 (𝑣 = 𝑤 → (𝑣𝑧) = (𝑤𝑧))
147 eqid 2813 . . . . . . . 8 (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) = (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))
148 ovex 6909 . . . . . . . 8 (𝑤𝑧) ∈ V
149146, 147, 148fvmpt 6506 . . . . . . 7 (𝑤 ∈ ran 𝑃 → ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))‘𝑤) = (𝑤𝑧))
150149adantl 469 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))‘𝑤) = (𝑤𝑧))
151 f1f 6319 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃⟶ℝ)
152 frn 6265 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃⟶ℝ → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ⊆ ℝ)
153143, 151, 1523syl 18 . . . . . . . . . 10 ((𝜑𝑧 ∈ ran 𝐺) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ⊆ ℝ)
154153sselda 3805 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝑦 ∈ ℝ)
15563adantr 468 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝑧 ∈ ℝ)
156154, 155readdcld 10357 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → (𝑦 + 𝑧) ∈ ℝ)
157105ad2antrr 708 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝐼:(ℝ × ℝ)⟶ℝ)
158157, 154, 155fovrnd 7039 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → (𝑦𝐼𝑧) ∈ ℝ)
159156, 158remulcld 10358 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℝ)
160159recnd 10356 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
161127, 128, 145, 150, 160fsumf1o 14680 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)))
162129sselda 3805 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑤 ∈ ℝ)
163162recnd 10356 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑤 ∈ ℂ)
164139adantr 468 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑧 ∈ ℂ)
165163, 164npcand 10684 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → ((𝑤𝑧) + 𝑧) = 𝑤)
166165oveq1d 6892 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → (((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)) = (𝑤 · ((𝑤𝑧)𝐼𝑧)))
167166sumeq2dv 14659 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑤 ∈ ran 𝑃(((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
168161, 167eqtrd 2847 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
16940ad2antrr 708 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (ran 𝐹 × ran 𝐺) ⊆ dom + )
170 simpr 473 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
171 simplr 776 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐺)
172 opelxpi 5355 . . . . . . . . . . . 12 ((𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺) → ⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺))
173170, 171, 172syl2anc 575 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺))
174 funfvima2 6721 . . . . . . . . . . . 12 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
17538, 174mpan 673 . . . . . . . . . . 11 ((ran 𝐹 × ran 𝐺) ⊆ dom + → (⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
176169, 173, 175sylc 65 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺)))
177 df-ov 6880 . . . . . . . . . 10 (𝑦 + 𝑧) = ( + ‘⟨𝑦, 𝑧⟩)
178176, 177, 483eltr4g 2909 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 + 𝑧) ∈ ran 𝑃)
17962adantlr 697 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
180179recnd 10356 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
181139adantr 468 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
182180, 181pncand 10681 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) − 𝑧) = 𝑦)
183182eqcomd 2819 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 = ((𝑦 + 𝑧) − 𝑧))
184 oveq1 6884 . . . . . . . . . 10 (𝑣 = (𝑦 + 𝑧) → (𝑣𝑧) = ((𝑦 + 𝑧) − 𝑧))
185184rspceeqv 3527 . . . . . . . . 9 (((𝑦 + 𝑧) ∈ ran 𝑃𝑦 = ((𝑦 + 𝑧) − 𝑧)) → ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
186178, 183, 185syl2anc 575 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
187186ralrimiva 3161 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → ∀𝑦 ∈ ran 𝐹𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
188 ssabral 3877 . . . . . . 7 (ran 𝐹 ⊆ {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)} ↔ ∀𝑦 ∈ ran 𝐹𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
189187, 188sylibr 225 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)})
190147rnmpt 5579 . . . . . 6 ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) = {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)}
191189, 190syl6sseqr 3856 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
19263adantr 468 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
193179, 192readdcld 10357 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 + 𝑧) ∈ ℝ)
194105ad2antrr 708 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
195194, 179, 192fovrnd 7039 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
196193, 195remulcld 10358 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℝ)
197196recnd 10356 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
198153ssdifd 3952 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹) ⊆ (ℝ ∖ ran 𝐹))
199198sselda 3805 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹)) → 𝑦 ∈ (ℝ ∖ ran 𝐹))
200 eldifi 3938 . . . . . . . . . . . . 13 (𝑦 ∈ (ℝ ∖ ran 𝐹) → 𝑦 ∈ ℝ)
201200ad2antrl 710 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → 𝑦 ∈ ℝ)
20263adantr 468 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → 𝑧 ∈ ℝ)
203 simprr 780 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
2041, 2, 102itg1addlem3 23685 . . . . . . . . . . . 12 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
205201, 202, 203, 204syl21anc 857 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
206 inss1 4036 . . . . . . . . . . . . . . 15 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦})
207 eldifn 3939 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (ℝ ∖ ran 𝐹) → ¬ 𝑦 ∈ ran 𝐹)
208207ad2antrl 710 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ 𝑦 ∈ ran 𝐹)
209 vex 3401 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
210 vex 3401 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
211210eliniseg 5711 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ V → (𝑣 ∈ (𝐹 “ {𝑦}) ↔ 𝑣𝐹𝑦))
212209, 211ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (𝐹 “ {𝑦}) ↔ 𝑣𝐹𝑦)
213210, 209brelrn 5564 . . . . . . . . . . . . . . . . . . 19 (𝑣𝐹𝑦𝑦 ∈ ran 𝐹)
214212, 213sylbi 208 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝐹 “ {𝑦}) → 𝑦 ∈ ran 𝐹)
215208, 214nsyl 137 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ 𝑣 ∈ (𝐹 “ {𝑦}))
216215pm2.21d 119 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑣 ∈ (𝐹 “ {𝑦}) → 𝑣 ∈ ∅))
217216ssrdv 3811 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝐹 “ {𝑦}) ⊆ ∅)
218206, 217syl5ss 3816 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ ∅)
219 ss0 4179 . . . . . . . . . . . . . 14 (((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ ∅ → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ∅)
220218, 219syl 17 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ∅)
221220fveq2d 6415 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = (vol‘∅))
222 0mbl 23526 . . . . . . . . . . . . . 14 ∅ ∈ dom vol
223 mblvol 23517 . . . . . . . . . . . . . 14 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
224222, 223ax-mp 5 . . . . . . . . . . . . 13 (vol‘∅) = (vol*‘∅)
225 ovol0 23480 . . . . . . . . . . . . 13 (vol*‘∅) = 0
226224, 225eqtri 2835 . . . . . . . . . . . 12 (vol‘∅) = 0
227221, 226syl6eq 2863 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = 0)
228205, 227eqtrd 2847 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦𝐼𝑧) = 0)
229228oveq2d 6893 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 + 𝑧) · 0))
230201, 202readdcld 10357 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦 + 𝑧) ∈ ℝ)
231230recnd 10356 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦 + 𝑧) ∈ ℂ)
232231mul01d 10523 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · 0) = 0)
233229, 232eqtrd 2847 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
234233expr 446 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ ∖ ran 𝐹)) → (¬ (𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0))
235 oveq12 6886 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦 + 𝑧) = (0 + 0))
236235, 98syl6eq 2863 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦 + 𝑧) = 0)
237 oveq12 6886 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦𝐼𝑧) = (0𝐼0))
238 0re 10330 . . . . . . . . . . 11 0 ∈ ℝ
239 iftrue 4292 . . . . . . . . . . . 12 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = 0)
240 c0ex 10322 . . . . . . . . . . . 12 0 ∈ V
241239, 102, 240ovmpt2a 7024 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → (0𝐼0) = 0)
242238, 238, 241mp2an 675 . . . . . . . . . 10 (0𝐼0) = 0
243237, 242syl6eq 2863 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦𝐼𝑧) = 0)
244236, 243oveq12d 6895 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (0 · 0))
245 0cn 10320 . . . . . . . . 9 0 ∈ ℂ
246245mul01i 10514 . . . . . . . 8 (0 · 0) = 0
247244, 246syl6eq 2863 . . . . . . 7 ((𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
248234, 247pm2.61d2 173 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ ∖ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
249199, 248syldan 581 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
250 f1ofo 6363 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
251145, 250syl 17 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
252 fofi 8494 . . . . . 6 ((ran 𝑃 ∈ Fin ∧ (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∈ Fin)
253128, 251, 252syl2anc 575 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∈ Fin)
254191, 197, 249, 253fsumss 14682 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
25533a1i 11 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → (ran 𝑃 ∖ {0}) ⊆ ran 𝑃)
256121an32s 634 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
257 dfin4 4076 . . . . . . . 8 (ran 𝑃 ∩ {0}) = (ran 𝑃 ∖ (ran 𝑃 ∖ {0}))
258 inss2 4037 . . . . . . . 8 (ran 𝑃 ∩ {0}) ⊆ {0}
259257, 258eqsstr3i 3840 . . . . . . 7 (ran 𝑃 ∖ (ran 𝑃 ∖ {0})) ⊆ {0}
260259sseli 3801 . . . . . 6 (𝑤 ∈ (ran 𝑃 ∖ (ran 𝑃 ∖ {0})) → 𝑤 ∈ {0})
261 elsni 4394 . . . . . . . . 9 (𝑤 ∈ {0} → 𝑤 = 0)
262261adantl 469 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑤 = 0)
263262oveq1d 6892 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = (0 · ((𝑤𝑧)𝐼𝑧)))
264105ad2antrr 708 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝐼:(ℝ × ℝ)⟶ℝ)
265262, 238syl6eqel 2900 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑤 ∈ ℝ)
26663adantr 468 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑧 ∈ ℝ)
267265, 266resubcld 10746 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤𝑧) ∈ ℝ)
268264, 267, 266fovrnd 7039 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → ((𝑤𝑧)𝐼𝑧) ∈ ℝ)
269268recnd 10356 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → ((𝑤𝑧)𝐼𝑧) ∈ ℂ)
270269mul02d 10522 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (0 · ((𝑤𝑧)𝐼𝑧)) = 0)
271263, 270eqtrd 2847 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = 0)
272260, 271sylan2 582 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ (ran 𝑃 ∖ (ran 𝑃 ∖ {0}))) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = 0)
273255, 256, 272, 128fsumss 14682 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
274168, 254, 2733eqtr4d 2857 . . 3 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
275274sumeq2dv 14659 . 2 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
276197anasss 454 . . 3 ((𝜑 ∧ (𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
2777, 5, 276fsumcom 14732 . 2 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
278124, 275, 2773eqtr2d 2853 1 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  {cab 2799  wne 2985  wral 3103  wrex 3104  Vcvv 3398  cdif 3773  cin 3775  wss 3776  c0 4123  ifcif 4286  {csn 4377  cop 4383   ciun 4719   class class class wbr 4851  cmpt 4930   × cxp 5316  ccnv 5317  dom cdm 5318  ran crn 5319  cres 5320  cima 5321  Fun wfun 6098   Fn wfn 6099  wf 6100  1-1wf1 6101  ontowfo 6102  1-1-ontowf1o 6103  cfv 6104  (class class class)co 6877  cmpt2 6879  𝑓 cof 7128  Fincfn 8195  cc 10222  cr 10223  0cc0 10224   + caddc 10227   · cmul 10229  cmin 10554  Σcsu 14642  vol*covol 23449  volcvol 23450  1citg1 23602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302  ax-addf 10303
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-disj 4820  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-of 7130  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-sup 8590  df-inf 8591  df-oi 8657  df-card 9051  df-cda 9278  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-n0 11563  df-z 11647  df-uz 11908  df-q 12011  df-rp 12050  df-xadd 12166  df-ioo 12400  df-ico 12402  df-icc 12403  df-fz 12553  df-fzo 12693  df-fl 12820  df-seq 13028  df-exp 13087  df-hash 13341  df-cj 14065  df-re 14066  df-im 14067  df-sqrt 14201  df-abs 14202  df-clim 14445  df-sum 14643  df-xmet 19950  df-met 19951  df-ovol 23451  df-vol 23452  df-mbf 23606  df-itg1 23607
This theorem is referenced by:  itg1addlem5  23687
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